The large sample property of the iterative generalized least squares estimation for hierarchical mixed effects model

doi:10.3868/s140-DDD-023-0023-x

Front. Math. China

2023, Vol. 18

Issue (5) : 327-339 https://doi.org/10.3868/s140-DDD-023-0023-x

The large sample property of the iterative generalized least squares estimation for hierarchical mixed effects model

Chunyu WANG¹, Maozai TIAN^1,^2,³(

)

¹. Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing 100872, China
². School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, China
³. Xinjiang Social & Economic Statistics Research Center, School of Statistics and Information, Xinjiang University of Finance and Economics, Urumqi 830012, China

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Abstract

In many fields, we need to deal with hierarchically structured data. For this kind of data, hierarchical mixed effects model can show the correlation of variables in the same level by establishing a model for regression coefficients. Due to the complexity of the random part in this model, seeking an effective method to estimate the covariance matrix is an appealing issue. Iterative generalized least squares estimation method was proposed by Goldstein in 1986 and was applied in special case of hierarchical model. In this paper, we extend the method to the general hierarchical mixed effects model, derive its expressions in detail and apply it to economic examples.

Keywords Hierarchical model iterative generalized least squares estimation variance-covariance components maximum likelihood estimation

Corresponding Author(s): Maozai TIAN

Online First Date: 27 December 2023 Issue Date: 11 January 2024

Cite this article:

Chunyu WANG,Maozai TIAN. The large sample property of the iterative generalized least squares estimation for hierarchical mixed effects model[J]. Front. Math. China, 2023, 18(5): 327-339.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/s140-DDD-023-0023-x
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I5/327

Fig.1 The scatter plots for model (2.1)

Regression coefficient(fixed part)	True values	Estimation of parameters	Standard deviation	Variance-covariance components (random component)	True values	Estimation of parameters	Standard deviation
$β 0$	10	9.954	0.113	$σ u 0 2$	1.2	1.249	0.182
$β 1$	4	3.981	0.098	$σ u 01$	0.5	0.540	0.124
				$σ u 1 2$	1	0.923	0.137
				$σ e 2$	1	0.988	0.029

Tab.1 Results of iterative generalized least squares estimation for model (2.1) (Number of iterations: 4)

Regression coefficient(fixed part)	True values	Estimation of parameters	Standard deviation	Variance components (random component)	True values	Estimation of parameters	Standard deviation
$β 00$	10	9.977	0.132	$σ u 0 2$	1.2	1.245	0.200
$β 01$	?3	?2.952	0.174	$σ u 01$	0.5	0.582	0.025
$β 10$	4	3.952	0.020	$σ u 1 2$	1	1.014	0.005
$β 11$	2	2.010	0.022	$σ e 2$	1	0.966	0.028

Tab.2 Results of iterative generalized least squares estimation for model (5.1) (Number of iterations: 5)

Regression coefficient (fixed part)	Estimation of parameters	Standard deviation	Variance-covariance components (random component)	Estimation of parameters	Standard deviation
$β 00$	9.495	1.191	$σ u 0 2$	9.181	3.717
$β 01$	?0.103	0.019	$σ u 01$	1.397	0.566
$β 10$	0.780	0.181	$σ u 1 2$	0.213	0.087
			$σ e 2$	1.032	0.112

Tab.3 Results of iterative generalized least squares estimation for model (6.3)

1	R A Bradley, J J Gart. The asymptotic properties of ML estimators when sampling from associate populations. Biometrika 1962; 49(1/2): 205–214
2	H Goldstein. Hierarchical mixed linear model analysis using iterative generalized least squares. Biometrika 1986; 73(1): 43–56
3	H Goldstein. Hierarchical Statistical Models, 4th ed. Chichester: John Wiley & Sons, 2011
4	H V Henderson, S R Searle. The vec-permutation matrix the vec operator and Kronecker products: A review. Linear Multilinear Algebra 1981; 9(4): 271–288
5	S Lele, M L Taper. A composite likelihood approach to (co)variance components estimation. J Statist Plann Inference 2002; 103(1/2): 117–135
6	S R SearleG CasellaC E McCulloch. Variance Components. Hoboken, NJ: John Wiley & Sons, 2006
7	T A B SnijdersR J Bosker. Hierarchical Analysis: An Introduction to Basic and Advanced Hierarchical Modeling. London Sage, 1999
8	M Z Tian. Higher hierarchical Quantile Regression Modeling Theory. Beijing: Science Press, 2015
9	S G WangJ H ShiS J YinM X Wu. Introduction to linear models. Beijing: Science Press, 2004
10	S G Wang, S J Yin. A new estimate of the parameters in linear mixed models. Sci Sin Math 2002; 32(5): 434–443

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